Problem 4 - Chemistry The Central Science In Si Units

Question

Imagine that the reaction \(2 \mathrm{CO}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{CO}_{2}(g)\) occurs in a container that has a piston that moves to maintain a constant pressure when the reaction occurs at constant temperature. Which of the following statements describes how the volume of the container changes due to the reaction: (a) the volume increases by \(50 \%,(\mathbf{b})\) the volume increases by \(33 \%\), (c) the volume remains constant, (d) the volume decreases by \(33 \%,(\mathbf{e})\) the volume decreases by \(50 \% .\)

Step-by-Step Solution

Answer

The volume decreases by 33%, option (d).

Step 1: Determine Initial Moles of Gas

Before the reaction occurs, calculate the total number of moles of gas present using the stoichiometry of the reaction. We start with 2 moles of CO and 1 mole of O₂, which sums up to 3 moles of gas initially.

Step 2: Determine Moles of Gas After Reaction

After the reaction has completed, all the reactants are converted to products. We have 2 moles of CO₂ as the products, which means there are now 2 moles of gas in the container.

Step 3: Calculate the Change in Moles

The number of moles of gas changed from 3 moles initially to 2 moles after the reaction is complete. This is a decrease in moles.

Step 4: Relate Moles of Gas to Volume Change

At constant temperature and pressure, the volume of a gas is directly proportional to the number of moles (According to Avogadro's Law). Since the moles decreased from 3 to 2, we can determine the percentage change in volume.

Step 5: Calculate Percentage Change in Volume

The volume change can be calculated as follows: \[ \text{Percentage change} = \left( \frac{\text{Final moles} - \text{Initial moles}}{\text{Initial moles}} \right) \times 100 \% = \left( \frac{2 - 3}{3} \right) \times 100 \% = -33.33 \% \] Thus, the volume decreases by approximately 33\%.

Key Concepts

Avogadro's Law

Avogadro's Law is a fundamental principle in chemistry that states that equal volumes of gases, at the same temperature and pressure, contain an equal number of molecules. This implies that the volume of a gas is directly proportional to the number of moles when pressure and temperature are held constant. In simpler terms, if you increase the amount of gas (in moles), the volume increases, and vice versa.
If we consider a container where a reaction is occurring while maintaining constant pressure, Avogadro's Law helps us understand how the volume will change based on the change in moles. For example:

• Increasing the number of moles would lead to an increase in volume.
• Decreasing the number of moles would cause the volume to decrease, assuming pressure and temperature remain the same.

Avogadro’s Law is pivotal in solving problems that relate to gas volume changes in chemical reactions, especially under conditions of constant temperature and pressure. It gives us a straightforward way to link chemical changes to physical changes like volume fluctuation.

Reaction Stoichiometry

Reaction stoichiometry involves using balanced chemical equations to calculate the relationships between reactants and products in a chemical reaction. It allows chemists to predict how much of each substance is consumed or produced. Stoichiometry is essential in determining the molar relationships that describe how entire reactions will proceed.
When analyzing the reaction \[2 \text{CO}(g) + \text{O}_2(g) \rightarrow 2 \text{CO}_2(g)\] we see that:

• 2 moles of CO react with 1 mole of O₂.
• This produces 2 moles of CO₂.

The stoichiometry of this reaction is evident in the relationships: the initial moles of reactants are 2 moles of CO and 1 mole of O₂, adding up to 3 moles. However, after the reaction, the number of moles is reduced to 2, as represented by only CO₂ in the products.
This clear relationship between reactants and products allows us to apply stoichiometry to further understand changes like volume adjustments in gas reactions.

Volume Change

In gas reactions, volume change can be calculated using principles like those described by Avogadro’s Law. When a reaction occurs under constant pressure, the volume change is directly connected to the change in the number of gas moles in the reaction.
For the example reaction \[2 \text{CO}(g) + \text{O}_2(g) \rightarrow 2 \text{CO}_2(g)\] we started with 3 moles of gas and ended with 2 moles. The drop in molar quantity from 3 to 2 indicates a decrease in volume.

• The percentage change in volume can be calculated as: \[ \text{Percentage change} = \left(\frac{2 - 3}{3}\right) \times 100\% = -33.33\% \]
• This means the volume of the container decreases by 33.33%, reflecting a loss in gas quantity inside.

Understanding volume changes is crucial for predicting how gas reactions behave under differing conditions, thus providing insights into industrial and laboratory processes.